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Questions tagged [calculus-of-variations]

This tag is for problems relating to the calculus of variations that deal with maximizing or minimizing functionals. This problem is a generalization of the problem of finding extrema of functions of several variables. In fact, these variables will themselves be functions and we will be finding extrema of “functions of functions” or functionals.

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Smoothest logpdf distribution

I want to solve the following optimization $$\min_{f\in C^2(\mathbb{R})}\int_{-\infty}^{\infty}(\frac{d^2}{dx^2}\log(f(x)))^2 dx$$ subject to the constraint that $$\int_{-\infty}^{\infty} f(x)dx = 1 $$...
JEK's user avatar
  • 77
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0 answers
48 views

Calculus of variations with linear Lagrangian and two optimisations

I am working on a two-part system, that jointly determines two functions $F$ and $G$. $F'(x) = f(x)$ and $G'(x) = g(x)$. $F$ and $G$ are both defined on $[0, 1]$, $F(0) = G(0) = 0$, they are both non-...
Ishan Kashyap Hazarika's user avatar
0 votes
0 answers
28 views

Regularity for computing the first variation

I am having a trouble understanding the regularity needed to compute the first variation for the Euler-Lagrange equation for the functional $$F(u) = \int f(u) dx$$ Suppose $u:U \to \mathbb{R}$ for ...
Morcus's user avatar
  • 595
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0 answers
20 views

Quasiconvexity implies rank-one convexity proof

I'm struggling to understand the proof of Proposition 5.3 in Rindler's 'Calculus of Variations'. I cannot follow why $u_j$ is $0$ on the boundary of $Q_n$. The next line states that $u_j$ converges ...
Fluffy Alpaca's user avatar
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0 answers
27 views

Regarding Order of a function.

While going through VI Arnold's MMCM, I came across the following definition on Pg 56. I will present the relevant part here ... and R(h,$\gamma$)=$O$($h^2$), in the sense that for |h|<$\epsilon$ ...
Aditya Krishna Panickar's user avatar
4 votes
2 answers
157 views

Not enough degrees of freedom in Euler Lagrange equation

I have the following minimization problem $$ \min_{p,r} \int_0^\infty (r^2 + p^2) dt $$ subject to $$ \dot r = - r + \dot p,\quad p(0)=p_0,\quad r(0)=r_0, \quad p(\infty)=0, \quad r(\infty)=0. $$ ...
Alexander Vigodner's user avatar
1 vote
1 answer
63 views

Result check of a functional derivative

For quantum chemistry problems, I need to understand functional derivative. I have begun to learn them a few days ago. So, my skill is very low. In a exercise, I tried to calculate the functional ...
Stef1611's user avatar
  • 157
1 vote
1 answer
58 views

Lagrange multiplier for coupling flux across subdomains

I am solving two pdes: (1) $\nabla\cdot(\kappa \nabla u)=0$ in $\Omega = \Omega_1\bigcup\Omega_2$ where $\Omega$ can be thought of as a unit square vertically divided into two equal halves. (2) $\frac{...
ihsel's user avatar
  • 11
2 votes
1 answer
64 views

Finding limit of spectral radius to size ratios on a sequence of symmetric matrixes increasing in dimensions

I came across this problem recently: let $r_n$ be the spectral radius of the $n\times n$ matrix $A_n$ defined by ${A_n}_{(i,j)}=n-|i-j|$, find $$\lim_{n\to\infty}\frac{r_n}{n^2}$$ Intuitively, I guess ...
zap kabosu's user avatar
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0 answers
49 views

Intuition abount countable compactness

In the context of direct method for calculus of variations, it is useful to introduce the notion of countably compact space, more explicitly: Def: Let $K$ topological space, $K$ is countably compact ...
Manuel Bonanno's user avatar
0 votes
1 answer
32 views

Shape optimization for a given positive function

Consider a positive function $f(x,y,z)$ in $\mathbb{R}^3$. Consider also a bounded region in the space, say a ball $B(0,R)$. I want to find $$ \min_{\Omega \subset B(0,R)} \int_\Omega f \text{ with }...
tommy1996q's user avatar
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2 votes
1 answer
95 views

Trouble understanding $\frac{\partial y'}{\partial y}$ [duplicate]

I want to find the equation of the curve that minimizes the distance between two points $P \equiv (x_1,y_1)$ and $Q \equiv (x_2,y_2)$. Thus, with the curve $\gamma(x)=(t,y(x))$, we have that the ...
baristocrona's user avatar
0 votes
1 answer
45 views

What is the functional derivative of a general composition of functions?

Let $F_{a,b} := f_a\circ f_{a+1} \circ\cdots \circ f_{b-1}\circ f_b$ and $F_N := F_{1,N}$, i.e. $$F_N(x) = f_1(f_2(\cdots(f_N(x)))\cdots).$$ Then what is $$\frac{\delta F_N(x)}{\delta f_k(y)}\quad\...
Tobias Kienzler's user avatar
1 vote
0 answers
28 views

Functional derivative of a function with nested integrals

I'm trying to solve a calculus of variations problem to find the cross-sectional area of a bar as a function of its length, which minimises its volume but has some fixed displacement at the free end. ...
Ben's user avatar
  • 53
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0 answers
73 views

Geodesics from Euler-Lagrange's equations

I am trying to solve the following problem Considering any Riemannian surface, find the geodesics' equations from Euler-Lagrange's equations. I worked as follows. Let $S=(S,g)$ be a Riemannian ...
user1255055's user avatar
0 votes
0 answers
18 views

Trying to understand orthogonality of boundary conditions for functionals of the form $\int_{p_0}^{p_1}f(x,y)\sqrt{1+y'^2}dx$ bounded between 2 curves

A question I had whilst reading section 15 of Fomin's "Calculus of Variations" (great book btw!!) The General Question: Among all smooth curves whose end points $p_0$,$p_1$ lie between two ...
PhysicsIsHard's user avatar
0 votes
0 answers
18 views

Linear and bilinear forms in Ritz method

I'm trying to understand an article about Calculus of Variations applied to Solid Mechanics (https://www.researchgate.net/publication/...
Anonymous's user avatar
1 vote
0 answers
40 views

Sign of the first eigenfunction of the Laplacian

I am trying to prove that the first eigenfunction of the Laplacian operator in an open domain $\Omega$ does not change sign and that the first eigenvalue $\lambda_1$ is simple (with Dirichlet-boundary ...
Gattsu's user avatar
  • 21
1 vote
0 answers
28 views

Lagrangian for catenary problem with non-uniform/external force

It is well known that the Lagrangian for the catenary (hanging wire) with uniform gravitational force is. $$\mathcal{L}=\mu g y\sqrt{1+y'}$$ Now Suppose that there is a nonuniform, possibly external ...
Leon Kim's user avatar
  • 525
5 votes
0 answers
209 views

Subdifferential set of total variation norm

I have been trying to understand the Gateaux differential (if it does exist) of the total variation norm ($\|\cdot\|_\text{TV}$) over the space of measure $M(\textit{X}, \mathbb{R}^k)$ where $\textit{...
supernova's user avatar
3 votes
3 answers
176 views

Variations of Dawson's function

I am studying Dawson's function : $\displaystyle F : x \mapsto e^{-x^2}\int_0^x e^{t^2} dt$. I would like to prove that $F$ attains a maximum at a certain value $x_0 \in (0,1)$, and is increasing over ...
Henry's user avatar
  • 93
1 vote
2 answers
102 views

Why does the partial derivative of $y'$ with respect to $y$ vanish? [closed]

I'm following Weinstock's Calculus of Variations. On page 25 it says: We have in this case $$f=\frac{d g}{d x}=\frac{\partial g}{\partial x}+\frac{\partial g}{\partial y}y',$$ so that the Euler-...
sensorer's user avatar
1 vote
1 answer
49 views

Functional derivative different when constraints are explicitly substituted vs. Lagrange multiplier method

Question (brief) Is the result the same if one 1) takes a functional derivative on a constrained subspace, or 2) takes a functional derivative using the method of Lagrange multipliers to encode ...
Lucas Myers's user avatar
4 votes
1 answer
140 views

Find the extremals of the functional $ F(y) = \int_{1}^{2} \left( (y'(x))^2 x^2 + y^2(x) \right) \ dx $. Made a correction.

Find the extremals of the functional $ F(y) = \int_{1}^{2} \left( (y'(x))^2 x^2 + y^2(x) \right) \ dx $ subject to the conditions $y(1) = 0$, $y(2) = 0$ and the additional condition $ \int_{1}^{2} y^2(...
user avatar
1 vote
0 answers
42 views

Variational problem with inequality constraint

I am trying to minimize the following functional: $$ \int_0^1 ( f'(t)^2 + g'(t)^2 - 2 \, r f'(t) g'(t) ) \, dt $$ over all pairs $(f, g)$ such that $f(t) \geq f_0(t)$ and $g (t) \geq g_0(t)$ for all $...
tsnao's user avatar
  • 320
1 vote
0 answers
30 views

Why is the mapping degree given by $\int_{\mathbb R^2} \mathrm{det}(u|\partial_1 u|\partial_2 u) \, \mathrm dx$

I am currently reading the paper „Existence of Two-Dimensional Skyrmions via the Concentration-Compactness Method“ by Lin and Yang and in there, they define the mapping degree of a function $u : \...
DarkViole7's user avatar
1 vote
1 answer
39 views

Parameter control problem derived via Dirichlet principle / variational formulation of Poisson equation / Lagrange multipliers

The below is a distilled-down version of a more involved problem I am looking at. Suppose for simplicity that $\Omega = B_R(0) \subset \mathbb{R}^2$ and $R$ is very large. Let us further define $f_s\,\...
Pink and Floyd's user avatar
7 votes
3 answers
138 views

Function satisfying $y(0)=1$ and $y(1)=4$ such that $(y')^2/y<4$ for all $x\in (0,1)$

I am trying to find a function satisfying $$ y\left(0\right) = 1\quad\mbox{and}\quad y\left(1\right) = 4\quad\mbox{such that}\quad {\left(y'\right)^{2} \over y} < 4\,\, \forall x \in \left(0,1\...
antonffm's user avatar
2 votes
1 answer
58 views

Shortest path on the surface of a cylinder between given points $A$ and $B$

Suppose you have the cylinder $ x^2 + y^2 = R^2 $ And points $A = (R, 0, 0)$ and $ B = (0, R, h) $. Find the parametric equation of the curve of shortest length connecting $A$ and $B$. My attempt: If ...
that's what it is's user avatar
0 votes
0 answers
19 views

Variational calculus with constraints of boundary conditions. How to take into account

I am looking for references to understand the following. I've recently solved the thin plate functional minimization subjected to interpolation constraints. The calculations I did are mostly here: ...
user8469759's user avatar
  • 5,317
2 votes
1 answer
52 views

Stationary action principle with a non-local action

Typical presentations of the Euler-Lagrange equations in field theory assume that the Lagrangian density is a function of the fields and their derivatives, so that the action $S$ is the integral over ...
Matt Dickau's user avatar
  • 2,289
1 vote
1 answer
44 views

variational problem

The variational problem is following: $$ J[u,x] = \int _{ t _{ 0 } } ^{ t _{ 1 } } u \left( x \left( t \right) , t \right) {\text{d}t}$$ So how to calculate the $\delta J$ ? I am new to this subject, ...
randolf's user avatar
  • 13
1 vote
0 answers
23 views

Strictly convex function for minimization problem

I have the following function $$F\left(x,\lambda,\zeta\right)=\sqrt{\frac{1+\zeta^2}{\lambda-\mu x}},$$ due to a minimization problem of the kind $\operatorname{min}\int_0^1 F(x,y(x),y'(x)) dx$, i ...
Gonzalo de Ulloa's user avatar
2 votes
0 answers
62 views

Euler-Lagrange equation in matrix form

The Euler-Lagrange equation seems fundamental, and mappings or vector functions are prevalent. Thus, I find it peculiar that I haven't seen so far a convenient matrix form of the E-L. See section &...
Zohar Levi's user avatar
1 vote
0 answers
59 views

Proof of Du Bois-Reymond Lemma using Riesz representation theorem

I’m working with this version of the fundamental calculus of variations lemma: If $f\in L^p(\mathbb{R}^n)$ and $\int f\phi dx = 0 $ for all $\phi \in C^\infty_c(\mathbb{R}^n)$, then $f=0$ a.e. My ...
Shiva's user avatar
  • 133
1 vote
1 answer
62 views

$\ddot x$ vs. $\dot x^2$

I'm working on a physics assignment and am having some trouble. I need to integrate $r\dot\theta^2$ with respect to $t$. However, my trouble lies in the definition of the upper-dot format. Given: $$ \...
Chaserix's user avatar
5 votes
3 answers
236 views
+50

Minimizing surface area of revolution for a fixed volume of revolution

Suppose I want an open-topped cup to have the capacity to hold a volume $V$ of liquid. I want to find a shape for my cup that minimises its surface area. My attempt I strongly suspect that the shape ...
Cristof012's user avatar
0 votes
1 answer
45 views

Analytical solution for minimizing a functional involving a double integral

The core of diffusion models in machine learning is to find a denoiser function to approximate the diffusion noise. However, under simple settings, I would like to know if the optimal "denoiser&...
obfish's user avatar
  • 165
0 votes
2 answers
93 views

Landaus&Lifshitz's derivation of Euler-Lagrange equations

Let $ L\colon\mathbb R\times \mathbb R\times \mathbb R\to \mathbb R $ be a function (a "Lagrangian"), and let $$ S[q] = \int_{t_1}^{t_2} L(q(t),\dot q(t),t)\,\mathrm dt $$ for any curve $ q\...
GeometriaDifferenziale's user avatar
0 votes
2 answers
97 views

Demonstrate that a straight line is the shortest curve that connects two points

I know that there is a simple geometric demonstration based on the triangle inequality, but I am looking for a demonstration based on calculus. I know that I can use Euler-Lagrange equations, but I am ...
Marco Altieri's user avatar
3 votes
1 answer
113 views

Singular extremal of a constrained variational problem

Consider the following constrained variational problem: $$\min_{u \in H^{1}(I)} \{\mathcal{F}(u) : u(\pm 1) = 1, \mathcal{G}(u) = 1/3 \},$$ where $I = [-1, 1] \subseteq \mathbb{R}, H^1 (I) := H^{1, 2}(...
Lorenzo Catani's user avatar
0 votes
1 answer
30 views

In variational calculus, why weight function equals the variation of the primary variable $w=\delta u$

In the book, "An introduction to FEM" by J.Reddy, p32, it is stated that in the weak formulation, the weight function has the meaning of the variation of the dependent variable, i.e. in the ...
Hosein Javanmardi's user avatar
2 votes
1 answer
99 views

Extremal of a functional with one variable endpoint

Consider the functional $$J[y] = \int \limits_{0}^{b} {\frac{\sqrt{y'^2 + 1}}{y} \text{d}x}$$ with $y(0) = 0$ and the other endpoint somewhere along the circle $(x-9)^2 + y^2 - 9 = 0$ (call this ...
JOlv's user avatar
  • 99
1 vote
1 answer
65 views

Finding the extremal of a non-linear, second order functional.

The question reads as follows: Consider the functional $$J[y] = \int \limits_{0}^{1} {(y'' ^2 - 2k y)} \text{d}x$$ where $k$ is a constant. Find the extremal of $J$ satisfying the conditions $y(0) = y'...
JOlv's user avatar
  • 99
1 vote
0 answers
56 views

Euler-Lagrange equation for the function of the integral

Can I derive an Euler-Lagrange equation for the following functional: $$ F[u] = \phi(\int L(u,u^{'},x) dx) $$ with constraint $$ \int G(u,u^{'},x) dx = 0 $$ The function $\phi()$ can be any convex ...
Circle Z's user avatar
1 vote
1 answer
76 views

General version of the fundamental lemma of calculus of variations

Let the measure $\mu$ on $\mathcal{B}(\mathbb{R}^n)$ be a general Borel measure. Let $f:\mathbb{R^n} \to \mathbb{R}$ be a locally integrable function and $\int f\varphi d\mu =0$ for any smooth ...
Rain's user avatar
  • 125
0 votes
0 answers
32 views

How to solve Lagrange optimal control problem whose cost is independent of control?

I would like to solve an optimal control problem of the form $$ \inf_{u : [0, 1] \to \mathbb{R}_+} \int_0^1 L(x(t), t) \, dt \qquad \text{subject to} \qquad \dot{x}(t) = u(t) $$ where $L(x, t)$ is a ...
Max Daniels's user avatar
2 votes
3 answers
120 views

Lemma relating to the Fundamental Lemma of the Calculus of Variation

Recently I started reading "A First Course in the Calculus of Variations" by Mark Kot. One of the exercises in that book requires the reader to prove the following lemma: Let $M(x)\in C[a,b]...
The Little E's user avatar
6 votes
2 answers
231 views

The Variational form of a biharmonic PDE

Suppose $\Omega \subset \mathbb{R}^d$ is a $C^{1,1}$ domain. Consider the biharmonic boundary value problem (BVP): $$ \begin{cases} \Delta^2 u = f \\ \nabla u \cdot \nu = g \\ u = u_D \end{cases} $$ ...
Mr. Proof's user avatar
  • 1,575
0 votes
0 answers
25 views

variational form of the functional

when reading some article, i have some problem to get the force $\mathbf{F}$ related to the energy functional from Hamilton's principle. Given the functional: $$ E=\frac{1}{2}k_{s}\int\left(\left|\...
rn4th's user avatar
  • 21

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